{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,14]],"date-time":"2026-03-14T00:03:34Z","timestamp":1773446614732,"version":"3.50.1"},"reference-count":16,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2013,10,28]],"date-time":"2013-10-28T00:00:00Z","timestamp":1382918400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["The Review of Symbolic Logic"],"published-print":{"date-parts":[[2013,12]]},"abstract":"<jats:p>It is widely accepted that a theory of truth for arithmetic should be consistent, but<jats:italic>\u03c9<\/jats:italic>-consistency is less frequently required. This paper argues that<jats:italic>\u03c9<\/jats:italic>-consistency is a highly desirable feature for such theories. The point has already been made for first-order languages, though the evidence is not entirely conclusive. We show that in the second-order case the consequence of adopting<jats:italic>\u03c9<\/jats:italic>-inconsistent truth theories for arithmetic is unsatisfiability. In order to bring out this point, well known<jats:italic>\u03c9<\/jats:italic>-inconsistent theories of truth are considered: the revision theory of nearly stable truth<jats:bold>T<\/jats:bold><jats:sup>#<\/jats:sup>and the classical theory of symmetric truth<jats:bold>FS<\/jats:bold>. Briefly, we present some conceptual problems with<jats:italic>\u03c9<\/jats:italic>-inconsistent theories, and demonstrate some technical results that support our criticisms of such theories.<\/jats:p>","DOI":"10.1017\/s1755020313000269","type":"journal-article","created":{"date-parts":[[2013,10,28]],"date-time":"2013-10-28T10:59:41Z","timestamp":1382957981000},"page":"733-741","source":"Crossref","is-referenced-by-count":8,"title":["NOTES ON<i>\u03c9<\/i>-INCONSISTENT THEORIES OF TRUTH IN SECOND-ORDER LANGUAGES"],"prefix":"10.1017","volume":"6","author":[{"given":"EDUARDO","family":"BARRIO","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"LAVINIA","family":"PICOLLO","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2013,10,28]]},"reference":[{"key":"S1755020313000269_ref15","volume-title":"Foundations Without Foundationalism: A Case for Second-Order Logic","author":"Shapiro","year":"1991"},{"key":"S1755020313000269_ref10","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511921049"},{"key":"S1755020313000269_ref9","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1305\/ndjfl\/1040511340","article-title":"A system of complete and consistent truth","volume":"35","author":"Halbach","year":"1994","journal-title":"Notre Dame Journal of Formal Logic"},{"key":"S1755020313000269_ref8","doi-asserted-by":"crossref","DOI":"10.7551\/mitpress\/5938.001.0001","volume-title":"The Revision Theory of Truth","author":"Gupta","year":"1993"},{"key":"S1755020313000269_ref7","doi-asserted-by":"publisher","DOI":"10.1007\/BF00302338"},{"key":"S1755020313000269_ref5","volume-title":"From Kant to Hilbert: A Source Book in the Foundations of Mathematics","author":"Ewald","year":"1996"},{"key":"S1755020313000269_ref3","doi-asserted-by":"publisher","DOI":"10.1007\/BF00249666"},{"key":"S1755020313000269_ref12","doi-asserted-by":"publisher","DOI":"10.1007\/BF00302339"},{"key":"S1755020313000269_ref13","doi-asserted-by":"publisher","DOI":"10.1111\/j.1747-9991.2007.00070.x"},{"key":"S1755020313000269_ref1","doi-asserted-by":"publisher","DOI":"10.1007\/s11225-010-9289-8"},{"key":"S1755020313000269_ref11","volume-title":"Deflationism and Paradox","author":"Halbach","year":"2005"},{"key":"S1755020313000269_ref16","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780195083439.001.0001","volume-title":"The Liar Speaks the Truth. 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